Composition and Inverses
Logarithmic Functions.
Week 5. Logarithms.
MA161/MA1161: Semester 1 Calculus.
Prof. Götz Pfeiffer
School of Mathematics, Statistics and Applied Mathematics
NUI Galway
October 3-4, 2016
Limits
Composition and Inverses
Logarithmic Functions.
Reminder: Assignments.
Problem Set 1 is now due at 5pm, Friday October 7.
Problem Set 2 is now online and due at 5pm, Friday October 21.
If you need assistance, avail of the support in SUMS, and in the
tutorials:
1. Monday at 18:00 in IT202,
2. Tuesday at 11:00 in IT206,
3. Tuesday at 18:00 in AC201,
4. Wednesday at 14:00 in AC214,
5. Wednesday at 18:00 in AM104,
6. Thursday at 13:00 in IT125G.
Limits
Composition and Inverses
Logarithmic Functions.
Composition of Functions.
It is easy to make new functions from old ones, by forming their
sum, difference or product, and you’ve see many examples of this
already:
A polynomial is a sum of multiples of power functions.
And a rational function is the quotient of two polynomials.
Another important operation of this type is called composition.
The composite, f ◦ g, of two functions f and g is defined by the rule
(f ◦ g)(x) = f(g(x)).
We say “f after g” for f ◦ g, and pronounce f(g(x)) as “f of g of x”. It
means that we first apply g to x and then apply f to the result: f after
g, for short.
Examples
1. Let f(x) = x2 and g(x) = sin(x). Find f ◦ g and g ◦ f.
2. Let f(x) = 3x + 4 and g(x) = 2x2 + 5x. Find f ◦ g and g ◦ f.
Limits
Composition and Inverses
Logarithmic Functions.
One-to-one Functions.
One-to-one Functions.
A function f : D → R is called a one-to-one function if it never takes
on the same value twice, that is
f(x1 ) 6= f(x2 ) whenever x1 6= x2 .
Examples
• Is f(x) = x3 one-to-one?
• Is f(x) = x2 one-to-one?
• A function is one-to-one if it passes the horizontal line test.
Limits
Composition and Inverses
Logarithmic Functions.
Inverse Functions.
Suppose f : D → C is a one-to-one function with range C. Then its
inverse function f−1 has domain C and range D and is defined by
f−1 (x) = y ⇔ f(y) = x, for any x ∈ C.
Find the inverse function of f(x) = x3 + 2.
Cancellation.
Suppose f : D → C is a one-to-one function with inverse f−1 . Then
• f−1 (f(x)) = x for every x ∈ D.
• f(f−1 (x)) = x for every x ∈ C.
• Do not confuse the inverse function f−1 (x) with the reciprocal
1
f(x)−1 = f(x)
.
• Not every function has an inverse.
Limits
Composition and Inverses
Logarithmic Functions.
Limits
Recall: Properties of The Exponential Function.
Some properties of f(x) = ax :
• f(0) = a0 = 1.
• if x = n ∈ N then
24
y = ( 21 )x
y = 2x
22
n
f(x) = a = a · a · · · a
(n factors).
20
• f(−x) = a−x = 1/ax .
18
√
• f(1/x) = a1/x = x a.
• If x = p/q
√ ∈ Q then
√
f(x) = q ap = ( q a)p .
16
14
y
12
10
Laws of Exponents
8
x+y
1. a
x y
=a a .
ax
2. a
= y.
a
3. axy = (ax )y .
4. (ab)x = ax bx .
6
x−y
4
2
x
−4
2
−2
−2
4
Composition and Inverses
Logarithmic Functions.
Limits
Logarithmic Functions.
[see Section 1.6 of the Book]
• Recall: The logarithmic function loga , with base a, is the
inverse of the exponential function ax , with base a:
y = loga x ⇔ ay = x.
• Consider the case a = 2.
4
y = log2 x
2
y
1
2
3
4
5
6
7
8
9
10
11
12
13
x
−2
• log2 x is the inverse of f(x) = 2x ; e.g., log2 (8) = 3 since 23 = 8.
• log2 (x) is the power into which to raise 2 in order to get x.
• log10 (x) is the power into which to raise 10 in order to get x; e.g.,
log10 (1 000 000) = 6.
• For n ∈ N, the number log10 (n) is, roughly, the number of
(decimal) digits of n.
Composition and Inverses
Logarithmic Functions.
Limits
Properties of Logarithmic Functions.
y = log2 x
4
y = ln x
2
y = log10 x
y
−1
1
2
3
4
5
6
7
8
9
10
11
12
13
x
−2
Logarithms and exponential functions are inverse to each other:
• loga (ax ) = x for all x ∈ R.
• aloga (x) = x for all x > 0.
Logarithm Laws
• loga (xy) = loga (x) + loga (y).
• loga (x/y) = loga (x) − loga (y).
• loga (xr ) = r loga (x).
Composition and Inverses
Logarithmic Functions.
Logarithms: Common Bases.
The most common bases for logarithms are:
• base a = 10, relates to decimal representation of numbers;
log10 10 = 1, log10 1000 = 3, log10 10n = n.
• base a = 2, relates to binary representation of numbers:
64 = 26 = (1 000 000)2 =⇒ log2 64 = 6.
• base e = 2.71828182845905 . . . , Euler’s number. The natural
logarithm is written as
ln x := loge x.
To convert from one base, b say, to another, a, use
loga (x) =
logb (x)
ln x
=
.
logb (a)
ln a
log3 (1000) = log10 (1000)/ log10 (3) = 3/0.4771 = 6.2877.
Limits
Composition and Inverses
Logarithmic Functions.
Applications of Logarithmic Functions.
1. (Logarithmic Scales.) If a function is given by the equation
f(x) = xn for some n, then loga (f(x)) = n loga (x), i.e., the graph
of loga (f(x)) against loga (x) is a straight line, with slope n.
2. (Bacterial Growth revisted.) Recall from earlier, that a population
of Cyanobacteria can double four times every day. If at time t = 0
the population is p(0) = 1, 000 cells, the population at time t,
measured in hours, can be modelled as
p(t) = 1000 × 2t/6 .
How many hours does it take for the population to reach
250, 000?
3. (Stewart, Sec 1.6, Q.57) A bacteria population starts with 100
bacteria and doubles every three hours. So the number of
bacteria after t hours can be modelled as f(t) = 100 × 2t/3 .
When will the population reach 50, 000?
Limits
Composition and Inverses
Logarithmic Functions.
Limits.
[Section 2.1 of the Book]
The notion of a limit is one of the most fundamental in Calculus, and
is the gateway to understanding the notion of a derivative. Here is
the key idea:
The idea of a limit
We say that the limit of a function f at the point p is L if we can
make f(x) as close to L as we would like, by taking x as close to p as
is necessary. We write this as
lim f(x) = L.
x→p
This description raises a few questions:
1. What does it mean to “take x as close to p as necessary”?
2. What does it mean to “make f(x) as close to L as we would like”?
Let’s look at how this works in a few concrete examples.
Limits
Composition and Inverses
Logarithmic Functions.
Limits
The Trigonometric Limit, For Example
• The limit of f(x) as x → p, if it exists, is determined by function
values f(x) for values of x close to p but different from p.
• Sometimes, f(p) is not even defined, as p is not in the domain
of f. Still, the limit of f(x) as x → p might exist.
Example
sin x
and the limit of f(x) as x → 0. We
x
cannot compute f(0) as 0 is not in the domain of f. But we can
compute the value of f(x) for any x as close to 0 as we like. Plotting
the points obtained in this way, yields the graph of f(x):
Consider the function f(x) =
1
0.5
y
y = sin x/x
x
−8
−7
−6
−5
−4
−3
−2
−1
This suggests that lim f(x) = 1.
x→0
1
2
3
4
5
6
7
8
Composition and Inverses
Logarithmic Functions.
More Examples.
Example
Suppose that f(x) represents the distance (measured in meters) you
have travelled at time x (measured in seconds).
To determine the average speed over the time from x0 to x, we
f(x) − f(x0 )
compute
.
x − x0
To determine your velocity, i.e., the instantaneous change of
f(x) − f(x0 )
distance, we need to compute lim
.
x→x0
x − x0
Example
Sometimes we care about the long-term behaviour of a function. For
example, if p(t) describes the size of the population of bacteria at
time t, we might want to know whether in the long term the population
dies out, reaches a stable level, or just keeps growing and
growing . . . Here we need to compute lim p(t).
t→∞
Limits
Composition and Inverses
Logarithmic Functions.
How to Estimate a Limit.
Suppose that we want to estimate lim f(x).
x→p
1. Make a table of function values f(x) for values of x near p.
Include points both to the left and to the right of p. (Avoid x = p.)
2. Check if these values tend to approach some fixed value.
3. If so (and if this value is the same on the left and on the right
of p), then you can take this value as an estimate for the limit L. If
not, the limit might not exist after all.
(Watch out for rounding errors when using a calculator . . . )
Examples
Estimate the limit . . .
x2 − 4
1. . . . of f(x) = 3
as x → 2.
x − 3x − 2
x
2. . . . of f(x) = 2 as x → 0.
x
|x|
3. . . . of f(x) =
as x → 0.
x
Limits
Composition and Inverses
Logarithmic Functions.
One-Sided Limits.
The last example shows that estimates on the left of p can suggest a
different limit than those on the right. This gives rise to the notion of
one-sided limits.
Left-Hand Limit and Right-Hand Limit.
We write
lim f(x) = L (or lim+ f(x) = L)
x→p−
x→p
if f(x) approaches L, as x approaches p from the left (or right). This
means that we can make f(x) as close to L as we would like by taking
values of x < p (or x > p) as close to p as necessary.
Example
lim
x→0−
|x|
|x|
= −1 and lim+
= +1.
x→0
x
x
Limits
Composition and Inverses
Logarithmic Functions.
2-Sided Limits.
The (two-sided) limit of f(x) at p exists and is equal to L if and only if
both one-sided limits of f(x) at p exist and are equal to L.
lim f(x) = L if and only if lim− f(x) = L and lim+ f(x) = L.
x→p
x→p
Example
Consider the function f : R → R defined as
x2 , x 6 1,
f(x) =
x, x > 1.
What is its limit as x → 1?
x→p
Limits
Composition and Inverses
Logarithmic Functions.
Limits
Exercises.
1. Simplify the following
expressions.
(i)
(ii)
(iii)
(iv)
e3 ln 4 .
log4 64 + log2 1024.
ln(2e−x/2 ) − ln 2 + x2 .
ln 81/ ln 3.
2. Write down the values of
f(x) =
cos x
x − π/2
for some values of x near
π/2. Then guess lim f(x).
x→π/2
3. Estimate the value of
√
x2 + 9 − 3
lim
.
x→0
x2
4. The graph of
1 + x, x < −1,
f(x) = 1 2
x > −1,
2x ,
is shown below. Calculate the
following limits:
(i) lim+ f(x), (ii) lim− f(x),
x→0
x→0
(iii) lim f(x), (iv) lim+ f(x),
x→0
x→1
(v) lim− f(x), (vi) lim f(x).
x→1
x→1
y
1
1
x